Smoothing Algorithms For Solving Several Classes Of Variational Inequality Problems

Smoothing algorithms are e?ective for solving various optimization prob-lems. The smoothing functions play important roles in smoothing methods.The qualification of the smoothing functions will a?ect the theoretical analysisof the algorithm and the practical numerical results. Therefore, it is a goodtopic to how to propose a valuable smoothing function. It is well known thatthe smoothing algorithms are globally convergent and locally superlinearlyconvergent under suitable assumptions. Thus, it is another topic to proposea smoothing algorithm which has better convergence properties. Moreover,several smoothing algorithms have been presented and obtained the finite ter-mination of the algorithms under the suitable conditions. This thesis presentssmoothing algorithms for solving several classes of variational inequality prob-lems. Details are as follows:Firstly, we propose a class of smoothing functions and give some ba-sic properties of the new smoothing functions. We investigate a smoothing-type algorithm for solving the monotone a?ne variational inequality problem(AVIP) where a testing procedure is embedded into our algorithm. Specially,we reformulate the AVIP as a system of parameterized smooth equations, anduse a smoothing algorithm to solve the smooth equations. Under the assump-tion that the solution set of the AVIP is nonempty, we show that the proposedalgorithm may find a maximally complementary solution to the monotoneAVIP in a finite number of iterations.Secondly, using a smoothing function, we reformulate the equality and in-equality constrained monotone a?ne variational inequality problems (GAVIPfor short) as a system of parameterized smooth equations, and then proposea smoothing algorithm for solving the GAVIP . we show that the algorithmcan find an exact solution of the GAVIP in a finite number of iterations underan assumption that the solution set of the GAVIP is nonempty and the gen- eralized Jacobian matrix is nonsingular. The numerical results show that ourtheoretical findings coincide with the practical results.Finally, based on the MCP-function, we introduce a class of generalizedsmoothing functions. Some basic properties of the new smoothing functionsare investigated. By using this class of smoothing functions, we reformulatethe box constrained variational inequality problem as a system of parame-terized smooth equations, and then propose a smoothing algorithm with anonmonotone line search to solve the box constrained variational inequalityproblem. The proposed algorithm is proved to be globally and locally super-linearly convergent under suitable assumptions.